DONETSK, UKRAINE—Ukrainian authorities vowed Saturday to restore control over the roiling eastern part of their country, slowly advancing on two key breakaway cities even as the Kremlin and its supporters in Ukraine said the violence demanded a response.

The military operations Saturday claimed at least 10 lives, medical officials said, a day after a conflagration in a trade union building killed dozens of pro-Russian activists in the port city of Odesa in the bloodiest day in Ukraine in nearly three months.

Ukraine’s acting president, Oleksandr Turchynov, declared a two-day national mourning period, as grieving residents of Odesa streamed to the burned-out building to lay flowers.

Thousands of mourners took to the streets of the city, which until the previous day had been largely untouched by the violence that has plagued eastern Ukraine for weeks.

The violence was an ominous development in the unfolding turmoil in Ukraine, because Odesa, more than 550 kilometres to the west, lies on a belt extending into Moldova that Russian President Vladimir Putin has noted was historically part of the Russian Empire.

Reporters in Odesa said the mourners were chanting, “Odesa is a Russian city.”

By day’s end, Ukrainian authorities in the east said they had encircled Slovyansk and retaken portions of Kramatorsk, including a television tower that had been seized by pro-Russian activists, although the core of the city appeared to remain in rebel hands.

But for every advance that the Ukrainian government made, it seemed to lose ground elsewhere. Angry pro-Russian crowds seized control of more government buildings in Donetsk, and pro-Russian forces in Luhansk, a city just 22 kilometres from the Russian frontier, vowed war on Kyiv, seizing weapons inside a military recruiting centre.

The violence prompted the Kremlin to warn it was weighing how to respond as thousands of its troops massed along the border. Putin has previously said he is prepared to intervene if the interests of compatriots in Ukraine come under threat.

“Thousands” of calls had come in the previous 24 hours, he said, although he offered no evidence.

In Kyiv, authorities said they would not relent in their efforts to repel separatists in eastern Ukraine but that police officers in Odesa might be held accountable for allowing the violence there.

On Friday, a pro-Ukrainian rally in the city attended by thousands of soccer fans was attacked by pro-Russian separatists, sparking hours of street battles and causing the deaths of three people.

Later that evening, a pro-Ukrainian mob attacked people in a pro-Russian encampment, sending them running into a nearby building, which the mob then set on fire with gasoline bombs. Many people were trapped inside.

At least 46 people died in the clashes, almost all of them in the blaze, authorities said Saturday.

On the ground in eastern Ukraine, Kyiv’s control Saturday appeared patchy. In Donetsk, masked men, some dressed in black, others in camouflage, smashed and looted two buildings that symbolized the Ukrainian state as hundreds of civilians chanted “Odesa will not be forgiven,” as well as “Russia” and “No to fascism.”